Introduction to random processes

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Introduction to random processes

Jean-Fran¸cois Delmas October 20, 2021

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A starter on measure theory and random variables

In this chapter, we present in Section 1.1 a basic tool kit in measure theory with in mind the applications to probability theory. In Section 1.2, we develop the corresponding integration and expectation. The presentation of this chapter follows closely [1], see also [2].

We use the following conventionN={0,1, . . .} is the set of non-negative integers, N^∗ = NT

(0,+∞), and for m < n ∈ N, we set Jm, nK = [m, n]T

N. We shall consider R = RS

{±∞}= [−∞,+∞], and fora, b∈R, we writea∨b= max(a, b),a⁺=a∨0 the positive part ofa, and a⁻= (−a)⁺ its negative part.

For two sets A⊂E, the function 1A defined on E taking values in R is equal to 1 onA and to 0 onE\A.

1.1 Measures and measurable functions

1.1.1 Measurable space

Let Ω be a set also called a space. A measure on a set Ω is a function which gives the “size”

of subsets of Ω. We shall see that, if one asks the measure to satisfy some natural additive properties, it is not always possible to define the measure of any subsets of Ω. For this reason, we shall consider families of sub-sets of Ω called σ-fields. We denote by P(Ω) ={A;A⊂Ω}

the set of all subsets of Ω.

Definition 1.1. A collection of subsets of Ω, F ⊂ P(Ω), is called aσ-field on Ω if:

(i) Ω∈ F;

(ii) A∈ F implies A^c∈ F;

(iii) if (Ai, i∈I) is a finite or countable collection of elements of F, then S

i∈IAi ∈ F. We call (Ω,F) a measurable space and a set A∈ F is said to be F-measurable.

When there is no ambiguity on theσ-field we shall simply say that A is measurable instead of F-measurable. In probability theory a measurable set is also called an event. Properties

1

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(i) and (ii) implies that∅is measurable. Notice thatP(Ω) and{∅,Ω}areσ-fields. The latter is called the trivialσ-field. When Ω is at most countable, unless otherwise specified, we shall consider the σ-field P(Ω).

Proposition 1.2. Let C ⊂ P(Ω). There exists a smallest σ-field onΩ which contains C.

The smallest σ-field which contains C is denoted by σ(C) and is also called the σ-field generated byC.

Proof. Let (F_j, j∈J) be the collection of all the σ-fields on Ω containingC. This collection is not empty as it contains P(Ω). It is left to the reader to check that T

j∈JF_j is a σ-field.

Clearly, this is the smallest σ-field containingC.

Remark 1.3. In this remark we give an explicit description of a σ-field generated by a finite number of sets. Let C={A₁, . . . , An}, with n∈N^∗, be a finite collection of subsets of Ω. It is elementary to check that F =S

I∈IC_I;I ⊂ P(J1, nK) , with C_I =T

i∈IA_iT

j6∈IA^c_j and I ⊂J1, nK, is a σ-field. Notice that CIT

CJ =∅ for I 6=J. Thus, the subsetsCI are atoms of F in the sense that ifB ∈ F, thenCIT

B is equal to CI or to∅.

We shall prove that σ(C) = F. Since by construction C_I ∈ σ(C) for all I ⊂ J1, nK, we deduce that F ⊂σ(C). On the other hand, for all i∈ J1, nK, we have Ai =S

I⊂J1,nK, i∈ICI. This gives that C ⊂ F, and thus σ(C)⊂ F. In conclusion, we get σ(C) =F. ♦ If F and H are σ-fields on Ω, we denote by F ∨ H = σ(FSH) the σ-field generated by F and H. More generally, if (F_i, i ∈ I) is a collection of σ-fields on Ω, we denote by W

i∈IF_i =σ(S

i∈IF_i) theσ-field generated by (F_i, i∈I).

We shall consider product of measurable spaces. If (A_i, i∈I) is a collection of sets, then its product is denoted byQ

i∈IA_i ={(ω_i, i∈I);ω_i∈A_i ∀i∈I}.

Definition 1.4. Let ((Ωi,F_i), i∈I)be a collection of measurable spaces. The productσ-field N

i∈IF_i on the product space Q

i∈IΩ_i is the σ-field generated by all the sets Q

i∈IA_i such Ai∈ F_i for all i∈I and Ai= Ωi for alli∈I but for a finite number of indices.

When all the measurable spaces (Ωi,F_i) are the same for all i ∈I, say (Ω,F), then we also write the product space Q

i∈IΩ_i = Ω^I and the product σ-field N

i∈IF_i =F^⊗I.

We recall a topological space (E,O) is a spaceE and a collection Oof subsets of E such that: ∅and E belongs toO, any (finite or infinite) union of elements ofO belongs toO, and the intersection of any finite number of elements of O belongs toO. The elements ofO are called the open sets, and O is called a topology onE. There is a very naturalσ-field on a topological space.

Definition 1.5. If (E,O) is a topological space, then the Borel σ-field, B(E) =σ(O), is the σ-field generated by all the open sets. An element ofB(E) is called a Borel set.

Usually the Borelσ-field on E is different fromP(E).

Remark 1.6. Since all the open subsets of R can be written as the union of a countable number of bounded open intervals, we deduce that the Borel σ-field is generated by all the intervals (a, b) fora < b. It is not trivial to exhibit a set which is not a Borel set; an example was provided by Vitali¹.

1J. Stern. ”Le probl`eme de la mesure.” S´eminaire Bourbaki 26 (1983-1984): 325-346. http://eudml.org/

doc/110033.

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1.1. MEASURES AND MEASURABLE FUNCTIONS 3 Similarly to the one dimensional case, as all the open sets of R^d can be written as a countable union of open rectangles, the Borel σ-field on R^d, d ≥ 1, is generated by all the rectangles Qd

i=1(a_i, b_i) witha_i < b_i for 1≤i≤d. In particular, we get that the Borelσ-field

of R^dis the product² of thedBorel σ-fields onR. ♦

1.1.2 Measures

We give in this section the definition and some properties of measures and probability measures.

Definition 1.7. Let (Ω,F) be a measurable space.

(i) A [0,+∞]-valued function µ defined on F is σ-additive if for all finite or countable collection (Ai, i∈I) of measurable pairwise disjoint sets, that is Ai ∈ F for all i ∈I and A_i∩A_j =∅ for alli6=j, we have:

µ [

i∈I

A_i

!

=X

i∈I

µ(A_i). (1.1)

(ii) A measure µ on(Ω,F) is a σ-additive [0,+∞]-valued function defined on F such that µ(∅) = 0. We call(Ω,F, µ)a measured space. A measurable setAisµ-null ifµ(A) = 0.

(iii) A measureµ on(Ω,F) isσ-finite if there exists a sequence of measurable sets (Ωn, n∈ N) such thatS

n∈NΩ_n= Ω and µ(Ω_n)<+∞ for alln∈N.

(iv) A probability measure P on (Ω,F) is a measure on (Ω,F) such that P(Ω) = 1. The measured space(Ω,F,P) is also called a probability space.

We refer to Section 8.1 for the construction of measures such as the Lebesgue measure, see Proposition 8.4 and Remark 8.6, and the product probability measure, see Proposition 8.7.

Example 1.8. We give some examples of measures (check these are indeed measures). Let Ω be a space.

(i) The counting measure Card is defined byA7→Card (A) forA⊂Ω, with Card (A) the cardinal of A. It isσ-finite if and only if Ω is at most countable.

(ii) Let ω∈Ω. The Dirac measure at ω, δω, is defined byA7→ δω(A) =1_A(ω) forA⊂Ω.

It is a probability measure.

(iii) The Bernoulli probability distribution with parameterp∈[0,1], PB(p), is a probability measure on (R,B(R)) given by P_B(p)= (1−p)δ0+pδ1.

2Let (E1,O1) and (E2,O2) be two topological spaces. LetC={O1×O2;O1 ∈ O1, O2∈ O1}be the set of product of open sets. By definition, B(E1)⊗ B(E2) is the σ-field generated byC. The product topology O1⊗ O2onE1×E2is defined as the smallest topology onE1×E2containingC. The Borelσ-field onE1×E2, B(E1×E2), is theσ-field generated byO1⊗O2. SinceC ⊂ O1⊗O2, we deduce thatB(E1)⊗B(E2)⊂ B(E1×E2).

SinceO1⊗ O2 is stable by infinite (even uncountable) union, it might happens that the previous inclusion is not an equality, see Theorem 4.44 p. 149 from C. Aliprantis and K. Border. Infinite Dimensional Analysis.

Springer, 2006.

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(iv) The Lebesgue measureλon (R,B(R)) is a measure characterized byλ([a, b]) =b−afor all a < b. In particular, any finite set or (byσ-additivity) any countable set isλ-null³. The Lebesgue measure is σ-finite.

4 Let us mention that assuming only the additivity property (that is I is assumed to be finite in (1.1)), instead of the stronger σ-additivity property, for the definition of measures⁴ leads to a substantially different and less efficient approach. We give elementary properties of measures.

Proposition 1.9. Let µbe a measure on (Ω,F). We have the following properties.

(i) Additivity: µ(A∪B) +µ(A∩B) =µ(A) +µ(B), for allA, B ∈ F.

(ii) Monotonicity: A⊂B implies µ(A)≤µ(B), for allA, B∈ F.

(iii) Monotone convergence: If (An, n∈N) is a sequence of elements of F such that An ⊂ An+1 for alln∈N, then, we have:

µ [

n∈N

A_n

!

= lim

n→∞µ(A_n).

(iv) If (A_i, i∈I) is a finite or countable collection of elements of F, then we have the inequalityµ S

i∈IAi

≤P

i∈Iµ(Ai). In particular a finite or countable union ofµ-null sets is µ-null.

Proof. We prove (i). The setsA∩B^c,A∩B andA^c∩B are measurable and pairwise disjoint.

Using the additivity property three times, we get:

µ(A∪B) +µ(A∩B) =µ(A∩B^c) + 2µ(A∩B) +µ(A^c∩B) =µ(A) +µ(B).

We prove (ii). AsA^c∩B∈ F, we get by additivity thatµ(B) =µ(A) +µ(A^c∩B). Then use that µ(A^c∩B)≥0, to conclude.

We prove (iii). We setB₀ =A₀ andB_n=A_n∩A^c_n−1for alln∈N^∗so thatS

n≤mB_n=A_m for all m∈N^∗ and S

n∈NBn =S

n∈NAn. The sets (Bn, n≥0) are measurable and pairwise disjoint. By σ-additivity, we get µ(Am) =µ(S

n≤mBn) =P

n≤mµ(Bn) andµ S

n∈NAn

= µ S

n∈NB_n

= P

n∈Nµ(B_n). Use the convergence of the partial sums P

n≤mµ(B_n), whose terms are non-negative, towards P

n∈Nµ(B_n) as m goes to infinity to conclude.

Property (iv) is a direct consequence of properties (i) and (iii).

We give a property for probability measures, which is deduced from (i) of Proposition 1.9.

Corollary 1.10. Let(Ω,F,P) be a probability space andA∈ F. We haveP(A^c) = 1−P(A).

3A setA⊂Ris negligible if there exists aλ-null set B such thatA ⊂B (notice thatA might not be a Borel set). LetNλ be the sets of negligible sets. The Lebesgueσ-field,B^λ(R), onRis theσ-field generated by the Borelσ-field andNλ. By construction, we haveB(R)⊂ B^λ(R)⊂ P(R). It can be proved that those two inclusions are strict.

4H. F¨ollmer and A. Schied. Stochastic finance. An introduction in discrete time. De Gruyter, 2011.

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1.1. MEASURES AND MEASURABLE FUNCTIONS 5 We end this section with the definition of independent events.

Definition 1.11. Let (Ω,F,P)be a probability space. The events (Ai, i∈I)are independent if for all finite subset J ⊂I, we have:

P





\

j∈J

Aj



=Y

j∈J

P(Aj).

The σ-fields (F_i, i∈I) are independent if for all A_i ∈ F_i ⊂ F, i∈I, the events (A_i, i∈I) are independent.

1.1.3 Characterization of probability measures

In this section, we prove that if two probability measures coincide on a sufficiently large family of events, then they are equal, see the main results of Corollaries 1.14 and 1.15. After introducing a λ-system (or monotone class), we prove the monotone class theorem.

Definition 1.12. A collection A of sub-sets ofΩ is a λ-system (or monotone class) if:

(i) Ω∈ A;

(ii) A, B∈ A and A⊂B imply B∩A^c∈ A;

(iii) if(An, n∈N) is an increasing sequence of elements of A, then we have S

n∈NAn∈ A.

Theorem 1.13 (Mononote class Theorem). Let C be a collection of sub-sets of Ω stable by finite intersection (also called a π-system). All λ-system (or monotone class) containing C also contains σ(C).

Proof. Notice thatP(Ω) isλ-system containingcc. LetAbe the intersection of allλ-systems containing C. It is easy to check that A is the smallest λ-system containing C. It is clear that A satisfies properties (i) and (ii) from Definition 1.1. To check that property (iii) from Definition 1.1 holds also, so that Ais a σ-field, it is enough, according to property (iii) from Definition 1.12, to check thatAis stable by finite union or equivalently by finite intersection, thanks to property (ii) of Definition 1.12.

For B ⊂ Ω, set A_B = {A ⊂ Ω;A∩B ∈ A}. Assume that B ∈ C. It is easy to check that A_B is a λ-system, asC is stable by finite intersection, and that it contains C and thus A. Therefore, for all B∈ C,A∈ A, we getA∈ A_B and thus A∩B ∈ A.

Assume now that B ∈ A. It is easy to check that A_B is a λ-system. According to the previous part, it containsC and thus A. In particular, for all B∈ A,A∈ A, we getA∈ A_B and thus A∩B ∈ A. We deduce that A is stable by finite intersection and is therefore a σ-field. To conclude, notice that A containsC and thus σ(C) also.

Corollary 1.14. Let P and P⁰ be two probability measures defined on a measurable space (Ω,F) Let C ⊂ F be a collection of events stable by finite intersection. If P(A) =P⁰(A) for allA∈ C, then we have P(B) =P⁰(B) for all B∈σ(C).

Proof. Notice that {A ∈ F;P(A) = P⁰(A)} is a λ-system. It contains C. By the monotone class theorem, it containsσ(C).

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The next corollary is an immediate consequence of Definition 1.5 and Corollary 1.14.

Corollary 1.15. Let (E,O) be a topological space. Two probability measures on (E,B(E)) which coincide on the open sets O are equal.

1.1.4 Measurable functions

Let (S,S) and (E,E) be two measurable spaces. Letf be a function defined onS and taking values in E. For A⊂E, we set {f ∈A}=f⁻¹(A) ={x∈S;f(x)∈A}. It is easy to check that for A⊂E and (Ai, i∈I) a collection of subsets ofE, we have:

f⁻¹(A^c) =f⁻¹(A)^c, f⁻¹ [

i∈I

Ai

!

=[

i∈I

f⁻¹(Ai) and f⁻¹ \

i∈I

Ai

!

=\

i∈I

f⁻¹(Ai).

(1.2) We deduce from the properties (1.2) the following elementary lemma.

Lemma 1.16. Let f be a function from S to E andE a σ-field on E. The collection of sets f⁻¹(A); A∈ E is a σ-field onS.

Theσ-field

f⁻¹(A);A∈ E , denoted by σ(f), is also called theσ-field generated byf. Definition 1.17. A function f defined on a space S and taking values in a space E is measurable from (S,S) to (E,E) if σ(f)⊂ S.

When there is no ambiguity on theσ-fieldsS and E, we simply say thatf is measurable.

Example 1.18. Let A ⊂ S. The function 1_A is measurable from (S,S) to (R,B(R)) if and

only if A is measurable asσ(1_A) ={∅, S, A, A^c}. 4

The next proposition is useful to prove that a function is measurable.

Proposition 1.19. LetC be a collection of subsets ofE which generates the σ-fieldE onE.

A function f from S to E is measurable from (S,S) to (E,E) if and only if for all A ∈ C, f⁻¹(A)∈ S.

Proof. We denote by G the σ-field generated by

f⁻¹(A), A∈ C . We have G ⊂ σ(f). It is easy to check that the collection

A∈E;f⁻¹(A)∈ G is a σ-field on E. It contains C and thus E. This implies that σ(f) ⊂ G and thus σ(f) = G. We conclude using Definition 1.17.

We deduce the following result, which is important in practice.

Corollary 1.20. A continuous function defined on a topological space and taking values in a topological space is measurable with respect to the Borel σ-fields.

The next proposition concerns function taking values in product spaces.

Proposition 1.21. Let (S,S) and ((Ei,E_i), i ∈ I) be measurable spaces. For all i ∈ I, let f_i be a function defined on S taking values in E_i and set f = (f_i, i∈I). The function f is measurable from (S,S) to (Q

i∈IE_i,N

i∈IE_i) if and only if for all i ∈ I, the function f_i is measurable from (S,S) to (Ei,E_i).

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1.1. MEASURES AND MEASURABLE FUNCTIONS 7 Proof. By definition, the σ-field N

i∈IE_i is generated by Q

i∈IAi with Ai ∈ E_i and for all i∈I but one,Ai=Ei. LetQ

i∈IAi be such a set. Assume it is not equal toQ

i∈IEi and let i₀ denote the only index such that A_i₀ 6=E_i₀. We havef⁻¹ Q

i∈IA_i

=f_i⁻¹

0 (A_i₀)∈ S. Thus iff is measurable so is fi0. The converse is a consequence of Proposition 1.19.

The proof of the next proposition is immediate.

Proposition 1.22. Let (Ω,F), (S,S), (E,E) be three measurable spaces, f a measurable function defined on Ω taking values in S and g a measurable function defined on S taking values inE. The composed function g◦f defined onΩ and taking values inE is measurable.

We shall consider functions taking values in R. The Borel σ-field on R, B(R), is by definition the σ-field generated by B(R), {+∞} and {−∞} or equivalently by the family ([−∞, a), a ∈ R). We say a function (resp. a sequence) is real-valued if it takes values in (resp. its elements belong to) R. With the convention 0· ∞ = 0, the product of two real- valued functions is always defined. The sum of two functions f and g taking values in Ris well defined if (f, g) does not take the values (+∞,−∞) or (−∞,+∞).

Corollary 1.23. Let f and g be real-valued measurable functions defined on the same measurable space. The functions f g, f ∨g = max(f, g) are measurable. If (f, g) does not take the values (+∞,−∞) and (−∞,+∞), then the function f+g is measurable.

Proof. TheR²-valued functions defined onR² by (x, y)7→xy, (x, y)7→x∨yand (x, y)7→(x+ y)1_{(x,y)∈

R²\{(−∞,+∞),(+∞,−∞)}} are continuous onR² and thus measurable on R² according to Corollary 1.20. Thus, they are also measurable onR². The corollary is then a consequence of Proposition 1.22.

If (a_n, n∈N) is a real-valued sequence then its lower and upper limit are defined by:

lim inf

n→∞ a_n= lim

n→∞inf{a_k, k≥n} and lim sup

n→∞

a_n= lim

n→∞sup{a_k, k≥n}

and they belong to R. Notice that:

lim sup

n→∞ an= lim sup

n→∞ a⁺_n −lim inf

n→∞ a⁻_n.

The sequence (an, n∈N) converges (inR) if lim infn→∞an= lim sup_n→∞anand this common value, denoted by limn→∞an, belongs to R.

The next proposition asserts in particular that the limit of measurable functions is measurable.

Proposition 1.24. Let (fn, n∈N)be a sequence of real-valued measurable functions defined on a measurable space (S,S). The functions lim sup_n→∞f_n and lim infn→∞f_n are measurable. The set of convergence of the sequence,{x∈S; lim sup_n→∞f_n(x) = lim infn→∞f_n(x)}, is measurable. In particular, if the sequence (fn, n∈N) converges, then its limit, denoted by limn→∞f_n, is also measurable.

Proof. Fora∈R, we have:

lim sup

n→∞ fn< a

= [

k∈N^∗

[

m∈N

\

n≥m

fn≤a− 1 k

.

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Since the functions fn are measurable, we deduce that {lim sup_n→∞fn< a} is also measurable. Since the σ-field B(R) is generated by [−∞, a) fora∈R, we deduce from Proposition 1.19 that lim sup_n→∞f_n is measurable. Since lim infn→∞f_n = −lim sup_n→∞(−f_n), we deduce that lim infn→∞fn is measurable.

Leth= lim sup_n→∞f_n−lim infn→∞f_n, with the convention +∞ − ∞= 0. The function h is measurable thanks to Corollary 1.23. Since the set of convergence is equal to h⁻¹({0}) and that {0} is a Borel set, we deduce that the set of convergence is measurable.

We end this section with a very useful result which completes Proposition 1.22.

Proposition 1.25. Let (Ω,F), (S,S)be measurable spaces, f a measurable function defined on Ω taking values is S and ϕ a measurable function from (Ω, σ(f)) to (R,B(R)). Then, there exists a real-valued measurable function g defined on S such that ϕ=g◦f.

Proof. By simplicity, we assume that ϕ takes its values in R instead of R. For all k ∈ Z, n ∈ N the sets A_n,k = ϕ⁻¹([k2⁻ⁿ,(k+ 1)2⁻ⁿ)) are σ(f)-measurable. Thus, for all n ∈ N, there exists a collection (Bn,k, k∈Z) of sets ofS pairwise disjoint such thatS

k∈ZBn,k =S, B_n,k ∈ S and f⁻¹(B_n,k) = A_n,k for all k ∈ Z. For all n ∈ N, the real-valued function g_n= 2⁻ⁿP

k∈Zk1_B_n,k defined onS is measurable, and we haveg_n◦f ≤ϕ≤g_n◦f+ 2⁻ⁿ⁰ for n ≥n0 ≥0. The function g = lim sup_n→∞gn is measurable according to Proposition 1.24, and we have g◦f ≤ϕ≤g◦f+ 2⁻ⁿ⁰ for alln0 ∈N. This implies thatg◦f =ϕ.

1.1.5 Probability distribution and random variables

We first start with the definition of the image measure (or push-forward measure) which is obtained by transferring a measure using a measurable function. The proof of the next Lemma is elementary and left to the reader.

Lemma 1.26. Let (E,E, µ) be a measured space, (S,S) a measurable space, and f a measurable function defined on E and taking values in S. We define the function µf on S by µ_f(A) =µ(f⁻¹(A)) for allA∈ S. Then µ_f is a measure on(S,S).

The measure µ_f is called the push-forward measure or image measure of µby f. In what follow, we consider a probability space (Ω,F,P).

Definition 1.27. Let(E,E) be a measurable space. AnE-valued random variableX defined onΩis a measurable function from (Ω,F) to(E,E). Its probability distribution or law is the image probability measure PX.

At some point we shall specify theσ-field F on Ω, and say that X isF-measurable.

We say twoE-valued random variables X and Y are equal in distribution, and we write X ^(d)= Y, if PX =PY. For A∈ E, we recall we write {X ∈A}={ω;X(ω)∈A}=X⁻¹(A).

Two random variables X andY defined on the same probability space are equal a.s., and we write X ^a.s.= Y, if P(X =Y) = 1. Notice that ifX and Y are equal a.s., then they have the same probability distribution.

Remark 1.28. LetX be a real-valued random variable. Its cumulative distribution function FX is defined by FX(x) = P(X ≤x) for all x ∈ R. It is easy to deduce from Exercise 9.1

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1.1. MEASURES AND MEASURABLE FUNCTIONS 9 that if X andY are real-valued random variables, then X and Y are equal in distribution if

and only if F_X =F_Y. ♦

The next lemma gives a characterization of the distribution of a family of random variables.

Lemma 1.29. Let ((Ei,E_i), i∈I) be a collection of measurable spaces andX = (Xi, i∈I) a random variable taking values in the product space Q

i∈IE_i endowed with the productσ-field.

The distribution of X is characterized by the family of distributions of (Xj, j ∈ J) where J runs over all finite subsets of I.

According to Proposition 1.21, in the above lemma theXjis anEj-valued random variable and its marginal probability distribution can be recovered from the distribution ofX as:

P(Xj ∈Aj) =P X∈Y

i∈I

Ai

!

with Ai =Ei for i6=j.

Proof. According to Definition 1.4, the productσ-field E on the product spaceE =Q

i∈IEi

is generated by the family C of product sets Q

i∈IAi such Ai ∈ E_i for alli∈I and Ai =Ei

for alli6∈J, withJ ⊂I finite. Notice then that PX(Q

i∈IA_i) =P(X_j ∈A_j forj∈J). Since C is stable by finite intersection, we deduce from the monotone class theorem, and more precisely Corollary 1.14, that the probability measurePX on E is uniquely characterized by P(X_j ∈A_j forj∈J) for all J finite subset of I and allA_j ∈ E_j forj∈J.

We first give the definition of a random variable independent from aσ-field.

Definition 1.30. A random variable X taking values in a measurable space (E,E) is independent from aσ-fieldH ⊂ F ifσ(X) andHare independent or equivalently if, for allA∈ E and B∈ H, the events {X∈A} and B are independent.

We now give the definition of independent random variables.

Definition 1.31. The random variables(Xi, i∈I)are independent if theσ-fields(σ(Xi), i∈ I) are independent. Equivalently, the random variables (X_i, i∈I) are independent if for all finite subset J ⊂I, all Aj ∈ E_j with j∈J, we have:

P(Xj ∈Aj for allj∈J) =Y

j∈J

P(Xj ∈Aj).

We deduce from this definition that if the marginal probability distributions Pi of all the random variables X_i for i ∈ I are known and if (X_i, i ∈ I) are independent, then the distribution of X is the product probability N

i∈IP_i introduced in Proposition 8.7.

We end this section with the Bernoulli scheme.

Theorem 1.32. Let (E,E,P) be a probability space. Let I be a set of indices. Then, there exists a probability space and a sequence (X_i, i∈I) of E-valued random variables defined on this probability space which are independent and with the same distribution probability P.

When P is the Bernoulli probability distribution and I =N^∗, then (X_n, n∈N^∗) is called a Bernoulli scheme.

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Proof. For i ∈ I, set Ωi = E, F_i = E and Pi = P. According to Proposition 8.7, we can consider the product space Ω =Q

i∈IΩi with the productσ-field and the product probability N

i∈IPi. For all i∈ I, we consider the random variable: X_i(ω) = ω_i where ω = (ω_i, i ∈I).

Using Definition 1.31, we deduce that the random variables (Xi, i∈I) are independent with the same probability distribution P.

1.2 Integration and expectation

Using the results from the integration theory of Sections 1.2.1 and 1.2.2, we introduce in Section 1.2.5 the expectation of real-valued or R^d-valued random variables and give some well known inequalities. We study the properties of the L^p spaces in Section 1.2.3 and prove the Fubini theorem in Section 1.2.4. In Section 1.2.6 we collect some further results on independence.

1.2.1 Integration: construction and properties

Let (S,S, µ) be a measured space. The set R is endowed with its Borel σ-field. We use the convention 0· ∞ = 0. A function f defined on S is simple if it is real-valued, measurable and if there exists n ∈ N^∗, α₁, . . . , α_n ∈ [0,+∞], A₁, . . . , A_n ∈ S such that we have the representation f = Pn

k=1α_k1_A_k. The integral of f with respect to µ, denoted by µ(f) or R f dµ orR

f(x)µ(dx), is defined by:

µ(f) =

n

X

k=1

αkµ(Ak)∈[0,+∞].

Lemma 1.33. Let f be a simple function defined on S. The integral µ(f) does not depend on the choice of its representation.

Proof. Consider two representations for f: f =Pn

k=1α_k1_A_k =Pm

`=1β_`1_B_`, with n, m∈N^∗ and A1, . . . , An, B1, . . . , Bm ∈ S. We shall prove that Pn

k=1αkµ(Ak) =Pm

`=1β`µ(B`).

According to Remark 1.3, there exits a finite family of measurable sets (CI, I ∈ P(J1, n+ mK)) such that C_IT

C_J = ∅ if I 6= J and for all k ∈ J1, nK and ` ∈ J1, mK there exists I_k⊂J1, nKandJ_` ⊂J1, mKsuch thatAk =S

I∈I_kCI and B` =S

I∈J_`CI. We deduce that:

f = X

I∈P(J1,n+mK) n

X

k=1

α_k1_{I∈I_k_}

!

1_C_I = X

I∈P(J1,n+mK) m

X

`=1

β_`1_{I∈J_`_}

! 1_C_I

and thus Pn

k=1αk1{I∈I_k}=Pm

`=1β`1{I∈J_`} for all I such thatCI 6=∅. We get:

n

X

k=1

α_kµ(A_k) =X

I n

X

k=1

α_k1_{I∈I_k_}

!

µ(C_I) =X

I m

X

`=1

β_`1_{I∈J_`_}

!

µ(C_I) =

m

X

`=1

β_`µ(B_`), where we used the additivity of µfor the first and third equalities. This ends the proof.

It is elementary to check that iff and g are simple functions, then we get:

µ(af+bg) =aµ(f) +bµ(g) fora, b∈[0,+∞) (linearity), (1.3)

f ≤g⇒µ(f)≤µ(g) (monotonicity). (1.4)

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1.2. INTEGRATION AND EXPECTATION 11 Definition 1.34. Let f be a [0,+∞]-valued measurable function defined on S. We define the integral of f with respect to the measure µby:

µ(f) = sup{µ(g);g simple such that 0≤g≤f}.

The next lemma gives a representation of µ(f) using that a non-negative measurable functionf is the non-decreasing limit of a sequence of simple functions. Such sequence exists.

Indeed, one can define forn∈N^∗ the simple functionfnby fn(x) = min(n,2⁻ⁿb2ⁿf(x)c) for x ∈ S with the convention b+∞c = +∞. Then, the functions (f_n, n ∈ N^∗) are measurable and their non-decreasing limit is f.

Lemma 1.35. Let f be a [0,+∞]-valued function defined on S and (fn, n ∈ N) a non- decreasing sequence of simple functions such that limn→∞f_n = f. Then, we have that limn→∞µ(f_n) =µ(f).

Proof. It is enough to prove that for all non-decreasing sequence of simple functions (f_n, n∈ N) and simple function g such that limn→∞f_n ≥ g, we have limn→∞µ(f_n) ≥ µ(g). We deduce from the proof of Lemma 1.33 that there exists a representation of g such that g =P_N

k=1α_k1_A_k and the measurable sets (A_k,1≤k≤N) are pairwise disjoint. Using this representation and the linearity, we see it is enough to consider the particular caseg=α1_A, with α∈[0,+∞], A∈ S andfn1A^c = 0 for all n∈N.

By monotonicity, the sequence (µ(fn), n ∈ N) is non-decreasing and thus limn→∞µ(fn) is well defined, taking values in [0,+∞].

The result is clear ifα= 0. We assume thatα >0. Letα⁰ ∈[0, α). Forn∈N, we consider the measurable setsBn={f_n≥α⁰}. The sequence (B_n, n∈N) is non-decreasing withA as limit because limn→∞f_n ≥ g. The monotone property for measure, see property (iii) from Proposition 1.9, implies that limn→∞µ(Bn) = µ(A). As µ(fn) ≥ α⁰µ(Bn), we deduce that limn→∞µ(fn)≥α⁰µ(A) and that limn→∞µ(fn)≥µ(g) asα⁰ ∈[0, α) is arbitrary.

Corollary 1.36. The linearity and monotonicity properties, see (1.3) and (1.4), also hold for [0,+∞]-valued measurable functions f and g defined onS.

Proof. Let (f_n, n∈N) and (g_n, n ∈N) be two non-decreasing sequences of simple functions converging respectively towards f and g. Let a, b∈ [0,+∞). The non-decreasing sequence (afn+bgn, n∈N) of simple functions converges towardsaf+bg. By linearity, we get:

µ(af+bg) = lim

n→∞µ(afn+bgn) =a lim

n→∞µ(fn) +b lim

n→∞µ(gn) =aµ(f) +bµ(g).

Assume f ≤ g. The non-decreasing sequence ((fn ∨gn), n ∈ N) of simple functions converges towards g. By monotonicity, we get µ(f) = limn→∞µ(f_n)≤limn→∞µ(f_n∨g_n) = µ(g).

Recall that for a functionf, we writef⁺=f ∨0 = max(f,0) and f⁻= (−f)⁺.

Definition 1.37. Letf be a real-valued measurable function defined onS. The integral of f with respect to the measure µ is well defined if min (µ(f⁺), µ(f⁻))<+∞ and it is given by:

µ(f) =µ(f⁺)−µ(f⁻).

The function f isµ-integrable if max (µ(f⁺), µ(f⁻))<+∞ (i.e. µ(|f|)<+∞).

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We also write µ(f) = R

f dµ = R

f(x) µ(dx). A property holds µ-almost everywhere (µ-a.e.) if it holds on a measurable setB such thatµ(B^c) = 0. Ifµis a probability measure, then one saysµ-almost surely (µ-a.s.) forµ-a.e.. We shall omitµand write a.e. or a.s. when there is no ambiguity on the measure.

Lemma 1.38. Let f ≥0 be a real-valued measurable function defined on S. We have:

µ(f) = 0⇐⇒f = 0 µ-a.e..

Proof. The equivalence is clear iff is simple.

Whenf is not simple, consider a non-decreasing sequence of simple (non-negative) functions (fn, n∈N) which converges towards f. As {f 6= 0} is the non-decreasing limit of the measurable sets {f_n6= 0},n∈N, we deduce from the monotonicity property of Proposition 1.9, that f = 0 a.e. if and only if fn = 0 a.e. for all n∈ N. We deduce from the first part of the proof that f = 0 a.e. if and only if µ(fn) = 0 for all n ∈ N. As (µ(fn), n ∈ N) is non-decreasing and converges towards µ(f), we get thatµ(f_n) = 0 for alln∈Nif and only ifµ(f) = 0. We deduce that f = 0 a.e. if and only ifµ(f) = 0.

The next corollary asserts that it is enough to knowf a.e. to compute its integral.

Corollary 1.39. Let f and gbe two real-valued measurable functions defined on S such that µ(f) and µ(g) are well defined. If a.e. f =g, then we have µ(f) =µ(g).

Proof. Assume first thatf ≥0 and g≥0. By hypothesis the measurable set A={f 6=g}is µ-null. We deduce that a.e. f1_A= 0 andg1_A= 0. This implies that µ(f1_A) =µ(g1_A) = 0.

By linearity, see Corollary 1.36, we get:

µ(f) =µ(f1_A^c) +µ(f1_A) =µ(g1_A^c) =µ(g1_A^c) +µ(g1_A) =µ(g).

To conclude notice thatf =ga.e. implies that f⁺ =g⁺ a.e. and f⁻ =g⁻ a.e. and then use the first part of the proof to conclude.

The relation f = g a.e. is an equivalence relation on the set of real-valued measurable functions defined on S. We shall identify a function f with its equivalent class of all measurable functions g such that µ-a.e., f = g. Notice that if f is µ-integrable, then µ-a.e.

|f|<+∞. In particular, if f and g areµ-integrable, we shall writef+g for any element of the equivalent class off1_{|f_|<+∞}+g1{|g|<+∞}. Using this remark, we conclude this section with the following immediate corollary.

Corollary 1.40. The linearity property, see (1.3) witha, b∈R, and the monotonicity property (1.4), where f ≤ g can be replaced by f ≤ g a.e., hold for real-valued measurable µ-integrable functions f and g defined on S.

We deduce that the set ofR-valued µ-integrable functions defined onS is a vector space.

The linearity property (1.3) holds also on the set of real-valued measurable functions h such that µ(h⁺) < +∞ and on the set of real-valued measurable functions h such that µ(h⁻)<+∞. The monotonicity property holds also for real-valued measurable functions f and g such thatµ(f) and µ(g) are well defined.

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1.2. INTEGRATION AND EXPECTATION 13 1.2.2 Integration: convergence theorems

The a.e. convergence for sequences of measurable functions introduced below is weaker than the simple convergence and adapted to the convergence of integrals. Let (S,S, µ) be a measured space.

Definition 1.41. Let (fn, n ∈N) be a sequence of real-valued measurable functions defined on S. The sequence converges a.e. if a.e. lim infn→∞fn = lim sup_n→∞fn. We denote by limn→∞f_n any element of the equivalent class of the measurable functions which are a.e.

equal to lim infn→∞fn.

Notice that Proposition 1.24 assures indeed that lim infn→∞ is measurable. We thus deduce the following corollary.

Corollary 1.42. If a sequence of real-valued measurable functions defined on S converges a.e., then its limit is measurable.

We now give the three main results on the convergence of integrals for sequence of converging functions.

Theorem 1.43 (Monotone convergence theorem). Let (f_n, n ∈ N) be a sequence of real- valued measurable functions defined onS such that for all n∈N, a.e. 0≤fn≤fn+1. Then, we have:

n→∞lim Z

fndµ= Z

n→∞lim fndµ.

Proof. The setA={x;f_n(x)<0 or f_n(x)> f_n+1(x) for some n∈N}is µ-null as countable union ofµ-null sets. Thus, we get that a.e. f_n=f_n1_A^c for all n∈N. Corollary 1.39 implies that, replacing fn by fn1A^c without loss of generality, it is enough to prove the theorem under the stronger conditions: for all n ∈ N, 0 ≤ f_n ≤ f_n+1. We set f = limn→∞f_n the non-decreasing (everywhere) limit of (f_n, n∈N).

For all n ∈ N, let (f_n,k, k ∈N) be a non-decreasing sequence of simple functions which converges towards f_n. We set g_n = max{f_j,n; 1 ≤ j ≤ n}. The non-decreasing sequence of simple functions (g_n, n ∈ N) converges to f and thus limn→∞R

g_n dµ = R

f dµ. By monotonicity, gn ≤ fn ≤ f implies R

gn dµ ≤ R

fn dµ ≤R

f dµ. Taking the limit, we get limn→∞R

f_ndµ=R f dµ.

The proof of the next corollary is left to the reader (hint: use the monotone convergence theorem to get theσ-additivity).

Corollary 1.44. Let f be a real-valued measurable function defined on S such that a.e.

f ≥0. Then the function f µdefined on S by f µ(A) =R

1Af dµ is a measure on (S,S).

We shall say that the measuref µhas densityf with respect to the reference measureµ.

Fatou’s lemma will be used for the proof of the dominated convergence theorem, but it is also interesting by itself.

Lemma 1.45 (Fatou’s lemma). Let (f_n, n ∈ N) be a sequence of real-valued measurable functions defined on S such that a.e. fn ≥ 0 for all n ∈ N. Then, we have the lower

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semi-continuity property:

lim inf

n→∞

Z

f_ndµ≥ Z

lim inf

n→∞ f_ndµ.

Proof. The function lim infn→∞fn is the non-decreasing limit of the sequence (gn, n ∈ N) with gn= infk≥nf_k. We get:

Z

lim inf

n→∞ fndµ= lim

n→∞

Z

gndµ≤ lim

n→∞inf

k≥n

Z

f_kdµ= lim inf

n→∞

Z

fndµ,

where we used the monotone convergence theorem for the first equality and the monotonicity property of the integral for the inequality.

The next theorem and the monotone convergence theorem are very useful to exchange integration and limit.

Theorem 1.46 (Dominated convergence theorem). Let f, g,(f_n, n∈ N) and (g_n, n∈ N) be real-valued measurable functions defined on S. We assume that a.e.: |f_n| ≤gn for all n∈N, f = limn→∞f_n and g = limn→∞g_n. We also assume that limn→∞R

g_n dµ = R

g dµ and R gdµ <+∞. Then, we have:

n→∞lim Z

fndµ= Z

n→∞lim fndµ.

Takingg_n=gfor alln∈Nin the above theorem gives the following result.

Corollary 1.47 (Lebesgue’s dominated convergence theorem). Let f, g and (fn, n ∈ N) be real-valued measurable functions defined on S. We assume that a.e.: |f_n| ≤g for all n∈N, f = limn→∞f_n and R

gdµ <+∞. Then, we have:

n→∞lim Z

fndµ= Z

n→∞lim fndµ.

Proof of Theorem 1.46. As a.e. |f| ≤ g and R

g dµ < +∞, we get that the function f is integrable. The functions g_n+f_n and g_n−f_n are a.e. non-negative. Fatou’s lemma with gn+fnand gn−fngives:

Z

gdµ+ Z

f dµ= Z

(g+f) dµ≤lim inf

n→∞

Z

(g_n+f_n) dµ= Z

gdµ+ lim inf

n→∞

Z

f_ndµ, Z

gdµ− Z

f dµ= Z

(g−f) dµ≤lim inf

n→∞

Z

(gn−fn) dµ= Z

gdµ−lim sup

n→∞

Z

fndµ.

Since R

g dµ is finite, we deduce from those inequalities that R

f dµ ≤lim infn→∞R f_n dµ and that lim sup_n→∞R

fn dµ ≤ R

f dµ. Thus, the sequence (R

fn dµ, n ∈ N) converges towardsR

f dµ.

We shall use the next Corollary in Chapter 5, which is a direct consequence of Fatou’s lemma and dominated convergence theorem.

Corollary 1.48. Let f, g,(f_n, n∈N) be real-valued measurable functions defined on S. We assume that a.e. f_n⁺ ≤ g for all n∈ N, f = limn→∞f_n and that R

g dµ < +∞. Then, we have that (µ(fn), n∈N) and µ(f) are well defined as well as:

lim sup

n→∞

Z

f_ndµ≤ Z

n→∞lim f_ndµ.

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1.2. INTEGRATION AND EXPECTATION 15 1.2.3 The L^p space

Let (S,S, µ) be a measured space. We start this section with very useful inequalities.

Proposition 1.49. Let f andg be two real-valued measurable functions defined onS.

• H¨older inequality. Letp, q∈(1,+∞) such that ¹_p+¹_q = 1. Assume that |f|^p and|g|^q are integrable. Thenf g is integrable and we have:

Z

|f g|dµ≤ Z

|f|^pdµ

1/p Z

|g|^qdµ 1/q

.

The H¨older inequality is an equality if and only if there exists c, c⁰∈[0,+∞) such that (c, c⁰)6= (0,0)and a.e. c|f|^p =c⁰|g|^q.

• Cauchy-Schwarz inequality. Assume that f² and g² are integrable. Then f g is integrable and we have:

Z

|f g|dµ≤ Z

f²dµ

1/2 Z g²dµ

1/2

.

Furthermore, we haveR

f gdµ= R

f²dµ1/2 R

g²dµ1/2

if and only there exist c, c⁰ ∈ [0,+∞) such that (c, c⁰)6= (0,0)and a.e. c f =c⁰g.

• Minkowski inequality. Let p ∈ [1,+∞). Assume that |f|^p and |g|^p are integrable.

We have:

Z

|f+g|^pdµ 1/p

≤ Z

|f|^pdµ 1/p

+ Z

|g|^pdµ 1/p

.

Proof. H¨older inequality. We recall the convention 0·+∞= 0. The Young inequality states that for a, b∈[0,+∞],p, q∈(1,+∞) such that ¹_p+¹_q = 1, we have:

ab≤ 1 pa^p+1

q b^q.

Indeed, this inequality is obvious ifaorb belongs to {0,+∞}. For a, b∈(0,+∞), using the convexity of the exponential function, we get:

ab= exp

log(a^p)

p +log(b^q) q

≤ 1

pexp (log(a^p)) + 1

q exp (log(b^q)) = 1

pa^p+1 qb^q. Ifµ(|f|^p) = 0 orµ(|g|^q) = 0, the H¨older inequality is trivially true as a.e. f g= 0 thanks to Lemma 1.38. If this is not the case, then integrating with respect toµin the Young inequality witha=|f|/µ(|f|^p)^1/pand b=|g|/µ(|g|^q)^1/q gives the result. Because of the strict convexity of the exponential, if aand bare finite, then the Young inequality is an equality if and only if a^p and b^q are equal. This implies that, if |f|^p and |g|^q are integrable, then the H¨older inequality is an equality if and only there exist c, c⁰ ∈ [0,+∞) such that (c, c⁰)6= (0,0) and a.e. c|f|^p =c⁰|g|^q.

The Cauchy-Schwarz inequality is the H¨older inequality with p = q = 2. If R

f gdµ = Rf²dµ1/2 R

g²dµ1/2

, then since R

f gdµ ≤ R

|f g|dµ, the equality holds also in the

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Cauchy-Schwarz inequality. Thus there exists c, c⁰ ∈ [0,+∞) such that (c, c⁰) 6= (0,0) and c|f|= c⁰|g|. Notice also that R

(|f g| −f g) dµ = 0. Then use Lemma 1.38 to conclude that a.e. |f g|=f g and thus a.e. c f =c⁰g.

Letp≥1. From the convexity of the function x7→ |x|^p, we get (a+b)^p ≤2^p−1(a^p+b^p) for alla, b∈[0,+∞]. We deduce that|f+g|^p is integrable. The casep= 1 of the Minkowski inequality comes from the triangular inequality in R. Let p > 1. We assume that R

|f + g|^pdµ >0, otherwise the inequality is trivial. Using H¨older inequality, we get:

Z

|f+g|^pdµ≤ Z

|f||f+g|^p−1dµ+ Z

|g||f+g|^p−1dµ

≤ Z

|f|^pdµ 1/p

+ Z

|g|^pdµ 1/p!

Z

|f +g|^pdµ

(p−1)/p

.

Dividing by R

|f +g|^pdµ(p−1)/p

gives the Minkowski inequality.

For p ∈ [1,+∞), let L^p((S,S, µ)) denote the set of R-valued measurable functions f defined on S such that R

|f|^pdµ < +∞. When there is no ambiguity on the underlying space, resp. space and measure, we shall simply writeL^p(µ), resp. L^p. Minkowski inequality and the linearity of the integral yield that L^p is a vector space and the map k·k_p from L^p to [0,+∞) defined by kfk_p = R

|f|^pdµ1/p

is a semi-norm (that is kf +gk_p ≤ kfk_p+kgk_p and kafk_p ≤ |a| kfk_p forf, g∈ L^p and a∈R). Notice that kfk_p = 0 implies that a.e. f = 0 thanks to Lemma 1.38. Recall that the relation “a.e. equal to” is an equivalence relation on the set of real-valued measurable functions defined onS. We deduce that the space (L^p,k·k_p), whereL^p is the spaceL^p quotiented by the equivalence relation “a.e. equal to”, is a normed vector space. We shall use the same notation for an element of L^p and for its equivalent class in L^p. If we need to stress the dependence of on the measureµ of the measured space (S,S, µ) we may writeL^p(µ) and even L^p(S,S, µ) for L^p.

The next proposition asserts that the normed vector space (L^p,k·k_p) is complete and, by definition, is a Banach space. We recall that a sequence (f_n, n ∈ N) of elements of L^p converges in L^p to a limit, say f, iff ∈L^p, and limn→∞kf_n−fk_p= 0.

Proposition 1.50. Let p ∈[1,+∞). The normed vector space (L^p,k·k_p) is complete. That is every Cauchy sequence of elements of L^p converges in L^p: if (f_n, n∈N) is a sequence of elements of L^p such that limmin(n,m)→∞kf_n−fmk_p = 0, then there exists f ∈L^p such that limn→∞kf_n−fk_p = 0.

Proof. Let (f_n, n∈N) be a Cauchy sequence of elements of L^p, that is f_n∈L^p for all n∈N and limmin(n,m)→∞kf_n−fmk_p = 0. Consider the sub-sequence (nk, k∈N) defined byn0 = 0 and fork≥1,nk= inf{m > n_k−1;kf_i−fjk_p ≤2^−k for alli≥m, j ≥m}. In particular, we have

f_n_k+1−f_n_k

p ≤2^−kfor allk≥1. Minkowski inequality and the monotone convergence theorem imply that

P

k∈N|f_n_k+1−fnk|

p <+∞ and thus P

k∈N|f_n_k+1−fnk|is a.e. finite.

The series with general term (f_n_k+1−f_n_k) is a.e. absolutely converging. By considering the convergence of the partial sums, we get that the sequence (f_n_k, k∈N) converges a.e. towards a limit, say f. This limit is a real-valued measurable function, thanks to Corollary 1.42. We deduce from Fatou lemma that:

kf_m−fk_p ≤lim inf

k→∞ kf_m−f_n_kk_p.

Introduction to random processes